It seems to me like Quidditch doesn't actually require any magical ability (unless that's what makes the brooms fly, but since different brooms have different speeds I assume they're closer to cars than bicycles). Thus, it'd be a great way for a squib to overcome their magical disability, find a well-paying wizard job, and earn the respect of their peers.
Is there any indication in canon of a squib who was able to play Quidditch professionally?
No. All signs point to squibs being incapable of using broomsticks, largely based on the Pottermore article describing the life and times of Angus Buchanan, bestselling author of My Life As A Squib.
Angus assisted his brothers in their playing of Quidditch, but his involvement was apparently limited to catching snitches from the ground
Angus’s early childhood had been spent dodging curses on an almost daily basis, which meant that he was surprisingly fast for a man of his size. He found his greatest pleasure and pride in athleticism, and soon became adept at the relatively new Muggle sport of rugby. Years of helping his siblings catch Golden Snitches in the back garden also made him a natural at cricket.
He was incapable of flying a broomstick to Hogwarts, relying on his brother to get him there
On the appointed day, his big brother Hamish took him to Hogwarts on the back of his broomstick, hoping against desperate hope that Angus would be allowed to stay once they got there, or that the school might be able to tease some magic out of him.
